The Stacks project

Lemma 30.11.3. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules.

  1. If $\mathcal{G}$ has property $(S_1)$, then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ has property $(S_1)$.

  2. If $\mathcal{G}$ has property $(S_2)$, then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ has property $(S_2)$.

Proof. Follows immediately from Lemma 30.11.2 and the definitions. $\square$

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